let me give you an algorithm
let m+3=a^3
then m=a^3-3
now put values of perfect cubes (its easy) in place of a^3.for each such value you will get an integral value of m.
Do the same for m^2 + 3m + 3
that is m^2 + 3m + 3-a^3=0
solve this and insert different value of a^3.this time every solution may not be integral.